If $\mathrm{M,N}$ are $3\times 2, 2 \times 3$ matrices such that $\mathrm{MN}=\pmatrix{8& 2 & -2\\2& 5& 4\\-2& 4&5}$, then $\mathrm{det(NM)}$ is?
($\mathrm{NM}$ is invertible.)
$\mathrm{det(MN)}$ must be (and is) zero. But how to find $\mathrm{det(NM)}$? Any hint?
Best Answer
Hint If $p \geq q$ and $M, N$ are $p \times q$ and $q \times p$ matrices, respectively, then the characteristic polynomials of $p, q$ are related by $$ p_{MN}(\lambda) = \lambda^{p - q} p_{NM}(\lambda) . $$